Problem: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{y^2 - 9}{y - 3}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = y$ $ b = \sqrt{9} = -3$ So we can rewrite the expression as: $r = \dfrac{({y} {-3})({y} + {3})} {y - 3} $ We can divide the numerator and denominator by $(y - 3)$ on condition that $y \neq 3$ Therefore $r = y + 3; y \neq 3$